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This question is about writing a paper in mathematics.

I am studying a (partial differential) equation and an algorithm to solve it approximately (finite element method). I need to report in my work a theorem on the approximation error of this algorithm. Such theorem already exists in the paper X: it is proved in a context that is more general than mine for some reasons (moving vs non-moving domain), and slightly less general than mine for other reasons (homogeneous vs non-homogeneous boundary conditions).

Should I report the adapted version of the proof from paper X, after of course appropriately citing it?

The modifications I had to do to the existing proof are mostly "trivial": my estimations are just longer and slightly more technically involved versions of those in X. In one point, however, I make a non-trivial (not so ingenious either) conceptual passage. I need a lot of context to report and prove this conceptual passage, so that rewriting the whole proof in my paper is not so much additional work, and it would be of course clearer.

Long story short, reporting the whole proof seems pedantic and may not be well received from the reviewers, whereas saying "The proof is as in [X]." feels lazy or not informative enough to me. But there may be more leftover space for reporting original content directly related to my publication!

Any hints?

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    I think there’s no way to guess what a given reviewer might find preferable in such a situation. If you put the proof in, a reviewer might ask you to take it out, and if you leave it out they might ask you to add it. (And if there is more than one reviewer, they might disagree with each other.) So, probably it’s best to just do what you feel would make for the best paper that best serves the community of people interested in the topic of the paper.
    – Dan Romik
    May 20, 2023 at 18:37
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    If you're a student, you should ask your advisor. There's not a right or wrong answer, and your advisor will be better equipped to give you guidance than someone on the internet who only has the information you've provided here. May 20, 2023 at 18:39
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    A pertinent question is: how long is the proof? I have done both things depending on the length of the proof. If it is less than half a page, adding in the adapted proof (explaining it needs to be slightly adapted to the situation) should be fine. If it's longer, it's probably best not to, and just state what modifications need to be made to the proof (if they really are easy modifications). May 20, 2023 at 18:51

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Synthesizing a few answer comments as well as a suggestion of my own:

  • Ask your adviser if you're a student. They're likely to have a better gut feeling on both the specific result and community expectations.
  • If you're writing up, probably not the worst things in the world to write up the proof, finish drafting the paper, then consider whether your target journal or similar papers have explicit or implicit length guidelines. Then know you can remove the proof if the draft's too long.
  • If the proof is essentially routine, a bit longish, but has a nontrivial element: Consider deferring it to an Appendix. This makes it extremely easy to cut if a reviewer doesn't think it's necessary.
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    I strongly agree with the appendix option in most cases. In the main text, state the modified theorem. Then something say something like, "To prove this theorem one can take the strategy in X used to prove a similar theorem, with modifications Y and Z (see appendix for a full proof)." This also means reviewers will read it, and can tell you to add it back to the main paper if you need to. If you omit it entirely they won't peer review the proof and this can slow things down if they want modifications to the proof in the next round May 20, 2023 at 21:23
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Your concern about it being pedantic and not well received is unfounded. It is a healthy attitude that since their result does not cover the case you need, you consider it your (unfortunate) duty to give a proof. It may be pedantic, but pedantic in a good way, nobody should reproach you for it.

In my opinion, "the proof is as in X" should be considered an option only if the proof is verbatim as in X. Even then, it's better to acknowledge something like "the result in X formally does not apply to our case, but the proof transfers verbatim." If the proof is applies mutatis mutandis, write "the proof works verbatim once one replaces all derivatives with finite differences." In other words, it is an option to describe a proof in a "perturbative way" - by referring to an existing proof in the literature and describing what has to be changed. But at some point it may get awkward, and it becomes cleaner to give a self-contained exposition. It's your call as an author.

It's good, of course, if the role of the proof is properly explained: "We follow closely [1], with modification that may be obvious for an experienced reader. Nevertheless, because (reasons), we chose to include the proof." Relegating the proof to appendix, as suggested, is also a good way to signal your view of its originality.

If you submit to a journal with a page limit and have to cut out the proof, it would be a service to the community to keep a longer version publicly available on arXiv or your webpage.

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    +1 I think think it's worthwhile to stress again that the suggestion in the last paragraph (at least keep the proof in the appendix for the arXiv version) is particularly good. Mathematics is too full of folklore results that have never been properly written down by anybody because "it's a simple modification of XYZ" or "it's only a combination of well-known arguments" - where "simple" and "well-known" actually means "simple and well-known to those who have been working in the field for many years". May 21, 2023 at 10:22

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